If you're asking how the third conjugation historically came to use the vowels i/u, it has to do with regular sound changes that affected the original Proto-Indo-European (PIE) endings.
The present indicative active paradigm for this verb in PIE, with its descendant paradigm in Latin, would look something like this:
1sg. *h₂eǵ-ō > agō
2sg. *h₂eǵ-e-si > agis
3sg. *h₂eǵ-e-ti > agit
1pl. *h₂eǵ-o-mos > agimus
2pl. *h₂eǵ-e-tes > agitis
3pl. *h₂eǵ-o-nti > agunt
In PIE, as you can see, the personal endings were preceded by a so-called "thematic vowel", either e or o (except in the 1sg.). This vowel then underwent the following changes in Latin:
- In non-initial open syllables, all vowels (basically; there are some exceptions) in Latin became i. This is the source of the i in agimus, agitis.
- e became i before a word-final consonant. This is what happened in agis, agit, where first the final -i of the PIE form was lost (as also in the 3pl.), leaving the s/t word-final.
- o became u before a nasal (again not absolutely always, but in many cases). This gives the u of agunt.